DOCSLIB.ORG

Introduction to Curvilinear Coordinates

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Curvilinear Coordinates Appendix B Introduction to Curvilinear Coordinates B.1 Definition of a Vector A vector, v, in three-dimensional space is represented in the most general form as the summation of three components, v1, v2 and v3, aligned with three “base” vectors, as follows: 3 1 2 3 i v = v g1 + v g2 + v g3 = ∑ v gi (B.1) i=1 where bold typeface denotes vector quantities and the base vectors, gi, can be non-orthogonal and do not have to be unit vectors as long as they are non-coplanar. The subscript i indicates a covariant quantity and the superscript i indicates a contravariant quantity, hence the above formula describes vector v as three contravariant components of the covariant base vectors. The Einstein summation convention only applies where one dummy index i is subscript and the other is superscript (summation does not apply over a repeated subscript i, so that for instance the metric tensor gii, discussed later, has 3 separate components). B.2 Transformation Properties of Covariant and Contravariant Ten- sors The subject of covariant and contravariant tensors is often introduced in tensor analysis text books by defining the behaviour of the two under transformation. The gradient of a scalar, φ, is given by the following expression in general non-orthogonal coordinates (ξ;η;ζ): ∂φ ∂φ ∂φ ∂φ ∇φ = g1 + g2 + g2 = gi (B.2) ∂ξ ∂η ∂ζ ∂ξi 155 156 APPENDIX B. Introduction to Curvilinear Coordinates If one defines another coordinate system ξ;η;ζ then components of the gradient can be expressed using the chain-rule: ∂φ ∂ξ j ∂φ = (B.3) ∂ξi ∂ξi ∂ξ j which can be written: j Ai = ai A j (B.4) where: j ∂φ j ∂ξ ∂φ Ai = a = A j = (B.5) ∂ξ j i ∂ξi ∂ξ j Tensors that satisfy this transformation are called covariant tensors and have lowered subscripts, as in Ai. To examine the transformation properties of a contravariant tensor, the vector dr is considered, as follows: dr = dξg1 + dηg2 + dζg3 (B.6) As before, if one defines another coordinate system ξ;η;ζ then components of the vector can be expressed using the chain-rule: ∂ξi dξi = dξ j (B.7) ∂ξ j This can be written: i i j A = b jA (B.8) where: ∂ξi Ai dξi bi = A j = dξ j (B.9) ≡ j ∂ξ j Tensors that transform according to Equation (B.8) are termed contravariant, and have raised indices. i B.3 Covariant and Contravariant Base Vectors, gi and g One can define a point in space by the position vector, r, using the familiar Cartesian coordinates, as follows: r = xˆi + yjˆ + zkˆ 1 2 3 = x e1 + x e2 + x e3 i = x ei (B.10) and, equally, one can define the unit vector in the x-direction, ˆi, as follows: ∂r ˆi = (B.11) ∂x i B.3. Covariant and Contravariant Base Vectors, gi and g 157 or, more generally: ∂r e = (B.12) i ∂xi The same point in space can be defined using a more general coordinate system: r = ξg1 + ηg2 + ζg3 1 2 3 = ξ g1 + ξ g2 + ξ g3 i = ξ gi (B.13) where: ∂r g = (B.14) i ∂ξi Equations (B.10) and (B.13) are equivalent. Using the chain rule, one can therefore express the covariant general base vectors gi in terms of the covariant Cartesian base vectors, ei, as follows: ∂r ∂r ∂ξ j = ∂xi ∂ξ j ∂xi ∂ξ j e = g (B.15) i ∂xi j and likewise: ∂r ∂r ∂x j = ∂ξi ∂x j ∂ξi ∂x j g = e (B.16) i ∂ξi j The covariant and contravariant base vectors are defined such that the scalar product of the covari- ant and contravariant base vectors is unity, i.e.: j gi g = 1 if i = j · = 0 if i = j 6 or: j j gi g = δ (B.17) · i j i j where δ δ δi j is the Kronecker delta. i ≡ ≡ In Equation (B.13), the vector r was expressed in terms of the covariant base vector gi. In a similar way, vector r can be written in terms of the contravariant base vector gi: i r = ξig (B.18) 158 APPENDIX B. Introduction to Curvilinear Coordinates where, following a similar analysis to that given for Equation (B.16): ∂r ∂r ∂x = j (B.19) ∂ξi ∂x j ∂ξi i j and since g = ∂r=∂ξi and e = ∂r=∂x j, the contravariant base vector is given by: ∂x gi = j e j (B.20) ∂ξi One can obtain the covariant and contravariant components from the scalar product of the vector, i r, and the corresponding base vectors (gi or g ), as follows: j j r gi = ξ jg gi = ξ jδ = ξi (B.21) · · i i j i j i i r g = ξ g j g = ξ δ = ξ (B.22) · · j δ j ξ ξ ξ j where i has substitution operator properties (i.e. it changes the component j to i, or from to ξi). Comparing Equations (B.18) and (B.21) one can also see that if the base vector is taken from the i right-hand-side to the left-hand-side of Equation (B.18), the superscript g becomes subscript gi. There is an alternative method to obtaining the contravariant base vector gi as a function of e j to that shown above. Returning to Equation (B.15), it was shown that: ∂ξ j e = g (B.23) k ∂xk j Taking the scalar product of both sides of this equation with gi: ∂ξ j ∂ξ j ∂ξi i i i ek g = g j g = δ = (B.24) · ∂xk · ∂xk j ∂xk Now, assuming that the contravariant base vector gi can be obtained from e j using a linear combination αi of factors j: i αi 1 αi 2 αi 3 αi j g = 1e + 2e + 3e = je (B.25) and taking the scalar product of both sides of this with ek: i i j i j i g ek = α e ek = α δ = α (B.26) · j · j k k i i k where g ek = ∂ξ =∂x from Equation (B.24) and: · ∂ξi αi = (B.27) j ∂x j i B.3. Covariant and Contravariant Base Vectors, gi and g 159 Finally, from Equation (B.25), one obtains: ∂ξi gi = e j (B.28) ∂x j which is the same result as Equation (B.20). 1 The vector product (g2 g3) has magnitude equal to the area of the rectangle with sides g2 and × g3, with direction nˆ normal to both g2 and g3. The scalar product (g1 nˆ) is equivalent to a distance in · the normal direction, thus the volume of the parallelepiped spanned by vectors g1, g2 and g3 is given by: ∆Vol = g1 (g2 g3) (B.30) · × The contravariant base vectors also satisfy: 1 1 2 1 3 1 g = (g2 g3) g = (g3 g1) g = (g1 g2) (B.31) ∆Vol × ∆Vol × ∆Vol × and similarly the covariant base vectors satisfy: 1 2 3 1 3 1 1 1 2 g1 = ∆ g g g2 = ∆ g g g3 = ∆ g g (B.32) Vol0 × Vol0 × Vol0 × 1 2 3 where ∆Vol0 = g g g represents the volume of the parallelepiped spanned by the contravariant 1 2· ×3 base vectors g , g and g . It is useful to note at this point that the covariant and contravariant rectangular Cartesian base vec- m tors are identical, e em. This is partly why covariant and contravariant tensors are not mentioned in ≡ most fluid mechanics text books which only deal with Cartesian tensors. The equivalence of covariant and contravariant Cartesian tensors is demonstrated by: 1 1 g = (g2 g3) (B.33) ∆Vol × which states that the contravariant g1 vector is perpendicular to the plane defined by the two covariant 1 vectors, g2 and g3. In Cartesian coordinates there is no distinction between g and g1 since the kˆ vector is orthogonal to the plane defined by the ˆi and jˆ vectors (i.e. the g1 vector is perpendicular to the plane defined by g2 and g3). 1The vector product is defined as: (g2 g3) = ( g2 g3 sinθ)nˆ (B.29) × j jj j where nˆ is the unit normal to vectors g2 and g3 and θ is the angle between the two g2 and g3 vectors. Since the area of a triangle with sides g2 and g3 is determined from (1=2 base height) which is equivalent to (1=2 g2 g3 sinθ), the × × × j j × j j magnitude of the cross product must be equal to the area of the rectangle with sides g2 and g3 (i.e. two triangles). 160 APPENDIX B. Introduction to Curvilinear Coordinates B.4 The Jacobian Matrix, [J] It has previously been shown (Equations B.16 and B.28) that the covariant and contravariant base i j vectors, gi and g , can be expressed in terms of the Cartesian base vectors, e j or e , as follows: ∂x j g = e (B.34) i ∂ξi j ∂ξi gi = e j (B.35) ∂x j The Jacobian matrix, [J], is defined as the matrix of coefficients appearing in Equation (B.34): xξ xη xζ ∂x j [J] = = η (B.36) ∂ξi 2 yξ y yζ 3 zξ zη zζ 6 7 4 5 where, for example, xξ ∂x=∂ξ and all components are contravariant, i.e.: ≡ x x1 ≡ y x2 ≡ z x3 ≡ ξ ξ1 ≡ η ξ2 ≡ ζ ξ3 ≡ B.5 Determinant of the Jacobian Matrix, J The Jacobian, J, is defined as the determinant of the Jacobian matrix: J = det[J] = xξ yηzζ yζzη xη yξzζ yζzξ + xζ yξzη yηzξ (B.37) − − − − It was noted earlier that the base vectors used to describe vector r in three-dimensional space should not be coplanar.
Load more

Recommended publications
  • A.7 Orthogonal Curvilinear Coordinates
    ~NSORS Orthogonal Curvilinear Coordinates 569 )osition ated by converting its components (but not the unit dyads) to spherical coordinates, and 1 r+r', integrating each over the two spherical angles (see Section A.7). The off-diagonal terms in Eq. (A.6-13) vanish, again due to the symmetry. (A.6-6) A.7 ORTHOGONAL CURVILINEAR COORDINATES (A.6-7) Enormous simplificatons are achieved in solving a partial differential equation if all boundaries in the problem correspond to coordinate surfaces, which are surfaces gener­ (A.6-8) ated by holding one coordinate constant and varying the other two. Accordingly, many special coordinate systems have been devised to solve problems in particular geometries. to func­ The most useful of these systems are orthogonal; that is, at any point in space the he usual vectors aligned with the three coordinate directions are mutually perpendicular. In gen­ st calcu­ eral, the variation of a single coordinate will generate a curve in space, rather than a :lrSe and straight line; hence the term curvilinear. In this section a general discussion of orthogo­ nal curvilinear systems is given first, and then the relationships for cylindrical and spher­ ical coordinates are derived as special cases. The presentation here closely follows that in Hildebrand (1976). ial prop­ Base Vectors ve obtain Let (Ul, U2' U3) represent the three coordinates in a general, curvilinear system, and let (A.6-9) ei be the unit vector that points in the direction of increasing ui• A curve produced by varying U;, with uj (j =1= i) held constant, will be referred to as a "u; curve." Although the base vectors are each of constant (unit) magnitude, the fact that a U; curve is not gener­ (A.6-1O) ally a straight line means that their direction is variable.
    [Show full text]
  • Area and Volume in Curvilinear Coordinates 1 Overview
    Area and Volume in curvilinear coordinates 1 Overview There are several reasons why we need to have a way of measuring ares and volumes in general relativity. the gravitational field of a point mass m will be singular at the location of the mass, so we typically compute the field of a mass density; i.e., the mass contained in a unit volume. If we are dealing with the Gauss law of electrodynamics (and there will be a similar law for gravity) then we need to compute the flux through an area. How should areas and volumes be defined in general metrics? Let us start in flat 3-dimensional space with Cartesian coordinates. Consider two vectors V;~ W~ . These vectors describe the sides of a parallelogram, whose area can be written as A~ = V~ × W~ (1) The cross-product has magnitude jA~j = jV~ jjW~ j sin θ (2) and is seen to equal the area of the parallelogram. The direction of the cross product also serves a useful purpose: it gives the normal direction to the area spanned by the vectors. Note that there are two normal directions to the area { pointing above the plane and below the plane { and the `right hand rule' for cross products is a convention that picks out one of these over the other. Thus there is a choice of convention in the choice of normal; we will see this fact again later. The cross product can be written in terms of a determinant 0 x^ y^ z^ 1 V~ × W~ = @ V x V y V z A (3) W x W y W z A determinant is computed by computing a product of numbers, taking one number from each row, while making sure that we also take only one number from each column.
    [Show full text]
  • Curvilinear Coordinate Systems
    Appendix A Curvilinear coordinate systems Results in the main text are given in one of the three most frequently used coordinate systems: Cartesian, cylindrical or spherical. Here, we provide the material necessary for formulation of the elasticity problems in an arbitrary curvilinear orthogonal coor- dinate system. We then specify the elasticity equations for four coordinate systems: elliptic cylindrical, bipolar cylindrical, toroidal and ellipsoidal. These coordinate systems (and a number of other, more complex systems) are discussed in detail in the book of Blokh (1964). The elasticity equations in general curvilinear coordinates are given in terms of either tensorial or physical components. Therefore, explanation of the relationship between these components and the necessary background are given in the text to follow. Curvilinear coordinates ~i (i = 1,2,3) can be specified by expressing them ~i = ~i(Xl,X2,X3) in terms of Cartesian coordinates Xi (i = 1,2,3). It is as- sumed that these functions have continuous first partial derivatives and Jacobian J = la~i/axjl =I- 0 everywhere. Surfaces ~i = const (i = 1,2,3) are called coordinate surfaces and pairs of these surfaces intersects along coordinate curves. If the coordinate surfaces intersect at an angle :rr /2, the curvilinear coordinate system is called orthogonal. The usual summation convention (summation over repeated indices from 1 to 3) is assumed, unless noted otherwise. Metric tensor In Cartesian coordinates Xi the square of the distance ds between two neighboring points in space is determined by ds 2 = dxi dXi Since we have ds2 = gkm d~k d~m where functions aXi aXi gkm = a~k a~m constitute components of the metric tensor.
    [Show full text]
  • Appendix a Orthogonal Curvilinear Coordinates
    Appendix A Orthogonal Curvilinear Coordinates Given a symmetry for a system under study, the calculations can be simplified by choosing, instead of a Cartesian coordinate system, another set of coordinates which takes advantage of that symmetry. For example calculations in spherical coordinates result easier for systems with spherical symmetry. In this chapter we will write the general form of the differential operators used in electrodynamics and then give their expressions in spherical and cylindrical coordi- nates.1 A.1 Orthogonal curvilinear coordinates A system of coordinates u1, u2, u3, can be defined so that the Cartesian coordinates x, y and z are known functions of the new coordinates: x = x(u1, u2, u3) y = y(u1, u2, u3) z = z(u1, u2, u3) (A.1) Systems of orthogonal curvilinear coordinates are defined as systems for which locally, nearby each point P(u1, u2, u3), the surfaces u1 = const, u2 = const, u3 = const are mutually orthogonal. An elementary cube, bounded by the surfaces u1 = const, u2 = const, u3 = const, as shown in Fig. A.1, will have its edges with lengths h1du1, h2du2, h3du3 where h1, h2, h3 are in general functions of u1, u2, u3. The length ds of the line- element OG, one diagonal of the cube, in Cartesian coordinates is: ds = (dx)2 + (dy)2 + (dz)2 1The orthogonal coordinates are presented with more details in J.A. Stratton, Electromagnetic The- ory, McGraw-Hill, 1941, where many other useful coordinate systems (elliptic, parabolic, bipolar, spheroidal, paraboloidal, ellipsoidal) are given. © Springer International Publishing Switzerland 2016 189 F. Lacava, Classical Electrodynamics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-39474-9 190 Appendix A: Orthogonal Curvilinear Coordinates Fig.
    [Show full text]
  • Curvilinear Coordinates
    UNM SUPPLEMENTAL BOOK DRAFT June 2004 Curvilinear Analysis in a Euclidean Space Presented in a framework and notation customized for students and professionals who are already familiar with Cartesian analysis in ordinary 3D physical engineering space. Rebecca M. Brannon Written by Rebecca Moss Brannon of Albuquerque NM, USA, in connection with adjunct teaching at the University of New Mexico. This document is the intellectual property of Rebecca Brannon. Copyright is reserved. June 2004 Table of contents PREFACE ................................................................................................................. iv Introduction ............................................................................................................. 1 Vector and Tensor Notation ........................................................................................................... 5 Homogeneous coordinates ............................................................................................................. 10 Curvilinear coordinates .................................................................................................................. 10 Difference between Affine (non-metric) and Metric spaces .......................................................... 11 Dual bases for irregular bases .............................................................................. 11 Modified summation convention ................................................................................................... 15 Important notation
    [Show full text]
  • Appendix a Relations Between Covariant and Contravariant Bases
    Appendix A Relations Between Covariant and Contravariant Bases The contravariant basis vector gk of the curvilinear coordinate of uk at the point P is perpendicular to the covariant bases gi and gj, as shown in Fig. A.1.This contravariant basis gk can be defined as or or a gk g  g ¼  ðA:1Þ i j oui ou j where a is the scalar factor; gk is the contravariant basis of the curvilinear coordinate of uk. Multiplying Eq. (A.1) by the covariant basis gk, the scalar factor a results in k k ðgi  gjÞ: gk ¼ aðg : gkÞ¼ad ¼ a ÂÃk ðA:2Þ ) a ¼ðgi  gjÞ : gk gi; gj; gk The scalar triple product of the covariant bases can be written as pffiffiffi a ¼ ½¼ðg1; g2; g3 g1  g2Þ : g3 ¼ g ¼ J ðA:3Þ where Jacobian J is the determinant of the covariant basis tensor G. The direction of the cross product vector in Eq. (A.1) is opposite if the dummy indices are interchanged with each other in Einstein summation convention. Therefore, the Levi-Civita permutation symbols (pseudo-tensor components) can be used in expression of the contravariant basis. ffiffiffi p k k g g ¼ J g ¼ðgi  gjÞ¼Àðgj  giÞ eijkðgi  gjÞ eijkðgi  gjÞ ðA:4Þ ) gk ¼ pffiffiffi ¼ g J where the Levi-Civita permutation symbols are defined by 8 <> þ1ifði; j; kÞ is an even permutation; eijk ¼ > À1ifði; j; kÞ is an odd permutation; : A:5 0ifi ¼ j; or i ¼ k; or j ¼ k ð Þ 1 , e ¼ ði À jÞÁðj À kÞÁðk À iÞ for i; j; k ¼ 1; 2; 3 ijk 2 H.
    [Show full text]
  • Del in Cylindrical and Spherical Coordinates - Wikipedia, the Free Encycl
    Del in cylindrical and spherical coordinates - Wikipedia, the free encycl... http://en.wikipedia.org/wiki/Nabla_in_cylindrical_and_spherical_coordi... Make a donation to Wikipedia and give the gift of knowledge! Del in cylindrical and spherical coordinates From Wikipedia, the free encyclopedia (Redirected from Nabla in cylindrical and spherical coordinates) This is a list of some vector calculus formulae of general use in working with standard coordinate systems. Table with the del operator in cylindrical and spherical coordinates Operation Cartesian coordinates (x,y,z) Cylindrical coordinates (ρ,φ,z) Spherical coordinates (r,θ,φ) Definition of coordinates A vector field Gradient Divergence Curl Laplace operator or Differential displacement Differential normal area Differential volume Non-trivial calculation rules: 1. (Laplacian) 2. 3. 4. (using Lagrange's formula for the cross product) 5. Remarks 1 of 2 10/20/2007 7:08 PM Del in cylindrical and spherical coordinates - Wikipedia, the free encycl... http://en.wikipedia.org/wiki/Nabla_in_cylindrical_and_spherical_coordi... This page uses standard physics notation; some (American mathematics) sources define φ as the angle from the z-axis instead of θ. The function atan2(y, x) is used instead of the mathematical function arctan(y/x) due to its domain and image. The classical arctan(y/x) has an image of (-π/2, +π/2), whereas atan2(y, x) is defined to have an image of (-π, π]. See also Orthogonal coordinates Curvilinear coordinates Vector fields in cylindrical and spherical coordinates Retrieved from "http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates" Categories: Vector calculus | Coordinate systems This page was last modified 11:17, 11 October 2007. All text is available under the terms of the GNU Free Documentation License.
    [Show full text]
  • Divergence and Curl Operators in Skew Coordinates
    International Journal of Modeling and Optimization, Vol. 5, No. 3, June 2015 Divergence and Curl Operators in Skew Coordinates Rajai S. Alassar and Mohammed A. Abushoshah where r is the position vector in any three dimensional space, Abstract—The divergence and curl operators appear in i.e. r r(,,) u1 u 2 u 3 . numerous differential equations governing engineering and physics problems. These operators, whose forms are well r Note that is a tangent vector to the ui curve where known in general orthogonal coordinates systems, assume ui different casts in different systems. In certain instances, one the other two coordinates variables remain constant. A unit needs to custom-make a coordinates system that my turn out to be skew (i.e. not orthogonal). Of course, the known formulas for tangent vector in this direction, therefore, is the divergence and curl operators in orthogonal coordinates are not useful in such cases, and one needs to derive their rr counterparts in skew systems. In this note, we derive two uuii r formulas for the divergence and curl operators in a general ei or hiei (3) coordinates system, whether orthogonal or not. These formulas r hi ui generalize the well known and widely used relations for ui orthogonal coordinates systems. In the process, we define an orthogonality indicator whose value ranges between zero and unity. The Divergence of a vector field over a control volume V bounded by the surface S , denoted by F , is defined by Index Terms—Coordinates systems, curl, divergence, Laplace, skew systems. F n dS F lim S (4) I.
    [Show full text]
  • Orthogonal Curvilinear Coordinates
    AN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES Overview Throughout the first few weeks of the semester, we have studied vector calculus using almost exclusively the familiar Cartesian x,y,z coordinate system. In your past math and physics classes, you have encountered other coordinate systems such as cylindri- cal polar coordinates and spherical coordinates. These three coordinate systems (Cartesian, cylindrical, spherical) are actually only a subset of a larger group of coordinate systems we call orthogonal coordinates. We are familiar that the unit vectors in the Cartesian system obey the relationship xi x j dij where d is the Kronecker delta. In other words, the dot product of any two unit vectors is 0 unless they are the same vector (in which case the dot product is one). This is the orthogonality property of vectors, and orthogonal coordinate systems are those in which all the unit vectors obey this property. We will learn general techniques for for translating vector operations into any orthogonal coordinate system, although we will be most concerned with the three systems used most frequently in basic physics (Cartesian, cylindrical polar, and spherical). We will start with very basic geometric properties of coordinate systems, and use those results as the basis for a more detailed analysis. Scale Factors Let' s start by considering a point in the x-y plane. We know we can describe the location of this point by the ordered pair (x,y) if we are using Cartesian coordinates, or by the ordered pair r,f) if we are using polar coordinates. You may also be familiar with the use of the symbols (r,q) for polar coordinates; either usage is fine, but I will try to be consistent in the use of (r,f) for plane polar coordinates, and (r,f,z) for cylindrical polar coordinates.
    [Show full text]
  • A Review: Operators Involving ∇In Curvilinear Orthogonal Coordinates
    A Review: Operators involving in Curvilinear Orthogonal Coordinates r and Vector Identities and Product Rules r Markus Selmke Student at the Universität Leipzig, IPSP,[email protected] 22.7.2006, updated 15.02.2008 Contents Abstract I 1 The Laplacian in Spherical Coordinates - The Tedious Way 1 2 The Angular Momentum Operators in Spherical Coordinates 4 3 Operators in Arbitrary Orthogonal Cuvilinear Coordinate Systems 5 r Summary of Important Formulas 11 4 Examples using the General Expressions 12 5 Proofs for the Product Rules 16 r 6 Gauss Divergence Theorem variants 18 References 19 Abstract This document provides easily understandable derivations for all important operators applicable to arbitrary orthog- 3 r onal coordinate systems in R (like spherical, cylindrical, elliptic, parabolic, hyperbolic,..). These include the laplacian (of a scalar function and a vector field), the gradient and its absolute value (of a scalar function), the divergence (of a vector field), the curl (of a vector field) and the convective operator (acting on a scalar function and a vector field) (Section 3). For the first two mentioned special coordinate systems the operators are calculated using these expressions along with some remarks on their simplifications in cartesian coordinates (Section 4). Alongside these calculations, some useful derivations are presented, including the volume element and the velocity components. The Laplacian as well as the absolute value of the gradient are also calculated the clumsy way (ab)using the chain-rule (Section 1). Using the same tedious way as in section 1, the angular momentum operators arising in quantum mechanics are derived for spherical coodinates (Section 2) only.
    [Show full text]
  • Tensor Analysis and Curvilinear Coordinates.Pdf
    Tensor Analysis and Curvilinear Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: May 19, 2016 Maple code is available upon request. Comments and errata are welcome. The material in this document is copyrighted by the author. The graphics look ratty in Windows Adobe PDF viewers when not scaled up, but look just fine in this excellent freeware viewer: http://www.tracker-software.com/pdf-xchange-products-comparison-chart . The table of contents has live links. Most PDF viewers provide these links as bookmarks on the left. Overview and Summary.........................................................................................................................7 1. The Transformation F: invertibility, coordinate lines, and level surfaces..................................12 Example 1: Polar coordinates (N=2)..................................................................................................13 Example 2: Spherical coordinates (N=3)...........................................................................................14 Cartesian Space and Quasi-Cartesian Space.......................................................................................15 Pictures A,B,C and D..........................................................................................................................16 Coordinate Lines.................................................................................................................................16 Example 1: Polar coordinates, coordinate lines
    [Show full text]
  • IX. ORTHOGONAL SYSTEMS Orthogonal Coordinate Systems
    IX. ORTHOGONAL SYSTEMS Orthogonal coordinate systems produce fewer additional terms in transformed partial differential equations, and thus reduce the amount of computation required. Also, as has been noted in Chapter V, severe departure from orthogonality will introduce truncation error in difference expressions. A general discussion of orthogonal systems on planes and curved surfaces is given in Ref. [42], and various generation procedures are surveyed in Ref. [42] and Ref. [1]. In numerical solutions, the concept of numerical orthogonality, i.e., that the off-diagonal metric coefficients vanish when evaluated numerically, is usually more important than strict analytical orthogonality, especially when the equations to be solved on the system are in the conservative law form. There are basically two types of orthogonal generation systems, those based on the construction of an orthogonal system from a non-orthogonal system, and those involving field solutions of partial differential equations. The first approach involves the construction of orthogonal trajectories on a given non-orthogonal system. Here one set of coordinate lines of the non-orthogonal system is retained, while the other set is replaced by lines emanating from a boundary and constructed by integration across the field so as to cross each line of the retained set orthogonally. Control of the line spacing is exercised through the generation of the non-orthogonal system and through the point distribution on the boundary from which the trajectories start. The point distributions on only three of the four boundaries can be specified. Several methods for the construction of orthogonal trajectories are discussed in Ref. [42] and Ref.
    [Show full text]